On the mod-$ell$ homology of the classifying space for commutativity

We study the mod-$ell$ homotopy type of classifying spaces for commutativity, $B(mathbb{Z}, G)$, at a prime $ell$. We show that the mod-$ell$ homology of $B(mathbb{Z}, G)$ depends on the mod-$ell$ homotopy type of $BG$ when $G$ is a compact connected Lie group, in the sense that a mod-$ell$ homology isomorphism $BG to BH$ for such groups induces a mod-$ell$ homology isomorphism $B(mathbb{Z}, G) to B(mathbb{Z}, H)$. In order to prove this result, we study a presentation of $B(mathbb{Z}, G)$ as a homotopy colimit over a topological poset of closed abelian subgroups, expanding on an idea of Adem and Gomez. We also study the relationship between the mod-$ell$ type of a Lie group $G(mathbb{C})$ and the locally finite group $G(bar{mathbb{F}}_p)$ where $G$ is a Chevalley group. We see that the naive analogue for $B(mathbb{Z}, G)$ of the celebrated Friedlander--Mislin result cannot hold, but we show that it does hold after taking the homotopy quotient of a $G$ action on $B(mathbb{Z}, G)$.